How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: MoeBlee on 5 Jul 2010 14:13 On Jul 3, 11:21 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > PA's axioms, together, would present an incomplete > picture about what we think of as the naturals Okay, and it does not vitiate anything I've said. I'm leaving out most of the rest of your arguments about this. Sorry, I just don't have time or interest to sort through them. > > So, in this way, certainty is put in terms that seems as unassailable > > as could be to me: IF there are any certain (non-logical) mathematical > > truths, then those of finitistic arithmetic are ones. I can't imagine > > that they are not certain, but even IF I could imagine them not > > certain, then I don't know what OTHER (non-logical) mathematical > > certain truths there can be. > > But why should mathematical truths be certain (i.e. absolute)? I didn't say that they should be. > > And even here, I don't disallow that > > there might be even more certain non-logical truths than those of > > finitistic arithmetic. It's just that presently I can't imagine them. > > I can't imagine what non-logical matter would qualify as certain if > > simple matching of strings of symbols is not certain. > > I probably miss something here but what does "matching of strings of > symbols" have to do with formula truth or with certainty of formula > truth? Could you elaborate? My time and interest is short now. I'd recommend reading more about Hilbert's notions. (Not that I claim to subscribe to Hilbert's notions without reservation; but rather that the notions are at least elaborated upon my him.) > > However, if someone doesn't even regard finitistic mathematics (such > > as results of PRA) - essentially just recognizing whether strings of > > symbols match or do not match - as "certain", "correct", whatever, > > then I admit that I can't see what basis for communication I would > > have with him or her. I don't know how we could even communicate if we > > couldn't agree that we can look at finite strings of symbols and check > > for matching. > > You've lost me here I'm afraid. Is PRA a formal system? Yes. > If so, why can't > you and the person communicate with each other the results of formal > proofs using rules of inference. You're mixed up about what I wrote. Moreover, you need to find out more about this basic subject matter. If you don't know that PRA is a formal system, then indeed you need to look into such things. > It's a curiosity that we've been talking about formal systems such as PA > and yet in this not-too-short post you've not mentioned inferences by FOL > rules of inference. Would there be reasons why you've not mentioned them > in this post? PRA is in FOL. MoeBlee
From: MoeBlee on 5 Jul 2010 14:21 On Jul 5, 7:05 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > "Jesse F. Hughes" <je...(a)phiwumbda.org> writes: > > > Aatu said PA is consistent, _period_, without any formal proof? > > There seems to be some confusion over my (perfectly standard as always) > take on these matters. Is there anything in my paraphrase several posts ago (in the full context I gave it) that is inaccurate? (Note: Just for the record, the particular quote above is not my own. However that quote is to be understood, for the record, I did not say that you hold there is no formal proof that PA is consistent, but rather that you hold PA is consistent on (for lack of better term I can think of right now) even more basic grounds than formal proof. > We can of course formalize this proof in any number of theories -- ACA, > ZFC, ... -- but this is just an incidental technical observation of no > immediate interest as far as consistency of PA is concerned. And that is what I have highlighted as to your view. MoeBlee
From: MoeBlee on 5 Jul 2010 14:30 On Jul 5, 11:12 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 3, 3:07 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> MoeBlee wrote: > >>> On Jul 3, 2:39 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> MoeBlee wrote: > FOL syntactical proof is well defined, and _does NOT require_ the > concept of "primitive arithmetic". I didn't say it does. > > And, in that regard, I have said all along that there is no finitistic > > proof of the consistency of PA. > > So you've agreed that there's no formal proof for PA's consistency NO!!! ONE LAST TIME: There ARE formal proofs that PA is consistent. But, as far as I know, there is no FINITISTIC formal proof that PA is consistent. As far as I can tell, formal proofs that PA is consistent carry no basis for believing that PA is consistent if one already doubted that PA is consistent. > and that > if you go by formal proof only then you don't have knowledge of PA's > consistency. I don't claim "knowledge" that PA is consistent. Rather, I find that there is good basis (aside from formal proof) to believe that PA is consistent. You did not at all understand my previous remarks about that. At this point, I may elect (as is always the case anyway) to allow your responses (including your plainly false statements (such as "So you've agreed that there's no formal proof for PA's consistency") to go without my remark. Life is just too short for trying to get through to you. MoeBlee
From: MoeBlee on 5 Jul 2010 14:47 On Jul 5, 11:36 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > The key question here is would the formal system containing "the formal > proof that PA is consistent" be _itself_ consistent, > according to Aatu? What Aatu claims about the consistency of the formal system doing the proving is not "the key question" for me. > You seemed to know what his views be on this matter, can you answer > this question? Yes. Aatu claims ZFC is consistent. He's written many posts on the subject. For even more on the subject, see Franzen's book MoeBlee
From: MoeBlee on 5 Jul 2010 14:59
On Jul 5, 11:47 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 5, 11:12 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> MoeBlee wrote: > >>> On Jul 3, 3:07 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> MoeBlee wrote: > >>>>> On Jul 3, 2:39 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>> MoeBlee wrote: > Do you MoeBlee agree that there's no formal proof that PA is > consistent in a consistent theory (formal system)? (Everything I say here pertains to formal.) I do not know that there is no consistent theory that proves PA is consistent. And with a theory that has a sole non-logical axiom that is a formalization of "PA is consistent", we do have a consistent theory that proves PA is consistent. But, AGAIN, I don't draw any epistemological import (basis for belief that PA is consistent) from such a thing. And aside from such theories as just mentioned, I do have basis to believe (without claiming certainty) that there exists a (non-trivial) consistent theory that proves PA is consistent. However, I do not take such a proof itself as a basis that one should believe PA is consistent if one already had strong doubts that PA is consistent. And I've explained about all of that in previous posts (either in this thread or in previous threads). Please read Franzen's book; as it would save a lot of typing for both of us. MoeBlee |